You may have noticed that I didn't list the modes in the usual order of: Ionian, Dorian, Phrygian, Lydian, etc.
I did it this way because I want you to think of these scales as either major or minor.

How can you tell if a scale or chord is major or minor? Good question, and there's a simple answer: A scale or chord is considered major, if it has a major third interval (two whole steps) between the first and third degrees.

A minor scale or chord has a minor third interval (one and a half steps) between the first and third degrees.

Refer to our chart above. Ionian, Lydian and Mixo-Lydian are major modes, because they have the major third interval. Since we have the modes grouped in this way, it's easy to see the differences between them:

Lydian is a major scale with a raised fourth.
Mixo-Lydian is a major scale with a lowered seventh.

Let's take a moment to think about what chords would go with these scales. Since the Lydian mode is a major scale with a raised fourth, wouldn't it make sense to use it over a major chord that contains a raised fourth? That would include Maj(#11), Maj7(#11). (If you are unfamiliar with chords and their spellings, see my Common Chord Formulas chart.)

How about our Mixo-Lydian scale? We said it was a major scale with a lowered seventh. Let's use it over a major chord with a lowered seventh, otherwise known as a dominant chord. The next time you see a G7, try playing a G Mixo-Lydian scale over it.

Let's go back to our chart. Notice that the Aeolian mode is the same as the natural minor scale. Let's use this scale to see the differences between the minor modes.

Dorian is the natural minor scale with a natural sixth.
Phrygian is the natural minor scale with a lowered second.
Locrian is the natural minor scale with a lowered second and fifth.

A word of warning: You may theoretically be able to play a Dorian scale over any minor chord, but let your ears be the final judge. It's really a matter of personal taste. I may play something that sounds a little "out" and think it's cool. It may just sound plain wrong to you. Listen to your ears; they're usually right.